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Schur's Lemma for Coupled Reducibility and Coupled Normality

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Schur's Lemma for Coupled Reducibility and Coupled Normality Swarthmore College Works Mathematics & Statistics Faculty Works Mathematics & Statistics 2019 Schur's Lemma For Coupled Reducibility And Coupled Normality D. Lahat C. Jutten Helene Shapiro Swarthmore College, [email protected] Follow this and additional works at: https://works.swarthmore.edu/fac-math-stat Part of the Mathematics Commons Let us know how access to these works benefits ouy Recommended Citation D. Lahat, C. Jutten, and Helene Shapiro. (2019). "Schur's Lemma For Coupled Reducibility And Coupled Normality". SIAM Journal On Matrix Analysis And Applications. Volume 40, Issue 3. DOI: 10.1137/ 18M1232462 https://works.swarthmore.edu/fac-math-stat/257 This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Mathematics & Statistics Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. Schur’s Lemma for Coupled Reducibility and Coupled Normality ∗Dana Lahat† Christian Jutten‡ Helene Shapiro§ December 3, 2018 Abstract Let = Aij i,j , where is an index set, be a doubly indexed A { } ∈I I family of matrices, where Aij is ni nj. For each i , let i be × ∈ I V an ni-dimensional vector space. We say is reducible in the coupled A sense if there exist subspaces, i i, with i = 0 for at least U ⊆ V U 6 { } one i , and i = i for at least one i, such that Aij( j) i ∈ I U 6 V U ⊆ U for all i, j. Let = Bij i,j also be a doubly indexed family of B { } ∈I matrices, where Bij is mi mj. For each i , let Xi be a matrix of × ∈I size ni mi. Suppose AijXj = XiBij for all i, j. We prove versions of × Schur’s Lemma for , satisfying coupled irreducibility conditions. A B We also consider a refinement of Schur’s Lemma for sets of normal matrices and prove corresponding versions for , satisfying coupled A B normality and coupled irreducibility conditions. ∗Funding: The work of D. Lahat and C. Jutten was supported by the project CHESS, 2012- ERC-AdG-320684. GIPSA-lab is a partner of the LabEx PERSYVAL-Lab (ANR11- LABX-0025). arXiv:1811.08467v2 [math.RA] 29 Nov 2018 †IRIT, Universit´ede Toulouse, CNRS, Toulouse, France ([email protected]). This work was carried out when D. Lahat was with GIPSA-lab, Grenoble, France. ‡Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France. C. Jutten is a senior member of Institut Univ. de France ([email protected] inp.fr). §Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081 ([email protected]). 1 1 Introduction Let K be a positive integer and let n1,...,nK and m1,...,mK be positive K integers. Consider two doubly indexed families of matrices, = Aij i,j=1 K A { K} and = Bij i,j=1, where Aij is ni nj and Bij is mi mj. Put N = i=1 ni B { K} × × and M = mi. Arrange the Aij’s into an N N matrix, A, withPAij in i=1 × block i, j ofPA. A A A K 11 12 ··· 1 A21 A22 A2K A = . ··· . AK AK AKK 1 2 ··· Similarly, form an M M matrix B with Bij in block i, j. Let Xi be an × ni mi matrix and form an N M matrix X with blocks X1,...,XK down the× diagonal and zero blocks elsewhere.× Thus, X1 0 0 0 0 X 0 ··· 0 2 ··· X = 0 0 X3 0 , . ··· . . 0 0 0 XK ··· where the 0 in position i, j represents an ni mi block of zeroes. Then × AX = XB if and only if AijXj = XiBij, (1) for all i, j = 1,...,K. We may rewrite AX = XB as AX XB = 0, a homogeneous Sylvester equation [13]. − We define several versions of coupled reducibility and prove corresponding versions of Schur’s Lemma [12] for pairs , . Imposing coupled irreducibil- A B ity constraints on and restricts the possible solutions, X1,...,XK, to the equations (1).A We alsoB discuss a refinement of Schur’s Lemma for nor- mal matrices, and prove corresponding versions for , satisfying coupled A B normality conditions. The system of coupled matrix equations (1) arises in a recent model for multiset data analysis [5, 8], called joint independent subspace analysis, or JISA. This model consists of K datasets, where each dataset is an un- known mixture of several latent stochastic multivariate signals. The blocks of A and B represent statistical links among these datasets. More specifi- cally, Aij and Bij represent statistical correlations among latent signals in the 2 ith and jth datasets. The multiset joint analysis framework, as opposed to the analysis of K distinct unrelated datasets, arises when a sufficient number of cross-correlations among datasets, i.e., Aij and Bij, i = j, are not zero. It turns out [6, 9, 10] that the identifiability and uniqueness6 of this model, in its simplest form, boil down to characterizing the set of solutions to the system of matrix equations (1), when the cross-correlations among the latent signals are subject to coupled (ir)reducibility. In other words, the coupled reducibility conditions that we introduce in this paper can be attributed with a physical meaning, and can be applied to real-world problems. We refer the reader to [6, 10] for further details on the JISA model, and on the derivation of (1). In this paper, we consider scenarios more general than those required to address the signal processing problem in [6, 10]. The analysis in Section 6, which focuses on coupled normal matrices, is inspired by the JISA model in [6, 10], in which the matrices A and B are Hermitian, and thus a spe- cial case of coupled normal. A limited version of some of the results in this manuscript was presented orally in, e.g., [7, 4, 1] and in a technical report [6]; however, they were never published. While motivated by the matrix equation, AX = XB, the main defi- nitions, theorems, and proofs do not require K to be finite. Thus, for a general index set, , we consider doubly indexed families, = Aij i,j I A { } ∈I and = Bij i,j . For the situation described above, = 1, 2,...,K . ∈I WeB use{ F }to denote the field of scalars. For some results,I { F can be} any field. For results involving unitary and normal matrices, F = C, the field of complex numbers. For each i , let ni and mi be positive integers, let i ∈ I V be an ni-dimensional vector space over F, and let i be an mi-dimensional ni W mi vector space over F. (Essentially, i = F and i = F .) For all i, j , V W ∈ I let Aij be an ni nj matrix and let Bij be mi mj. View Aij as a linear × × transformation from j to i, and Bij as a linear transformation from j V V W to i. For each i , let Xi be an ni mi matrix; view Xi as a linear W ∈ I × transformation from i to i. We are interested in families , satisfying the equations (1) forW all i, j V . A B ∈ I For some results, all of the ni’s will be equal, and all of the mi’s will be equal. In this case, we use n for the common value of the ni’s and m for the n m common value of the mi’s. We then set = F and = F . Each Xi is then n m. For = 1,...,K , we haveV N = Kn, andW M = Km. All of × I { } the blocks Aij in A are n n, while all of the blocks Bij in B are m m. × × Section 2 reviews the usual matrix version of Schur’s Lemma and its proof. Section 3 defines coupled reducibility and two restricted versions, 3 called proper and strong reducibility. Section 4 states and proves versions of Schur’s Lemma for coupled reducibility and proper reducibility, Theorem 4.2. In section 5, we define some graphs associated with the pair , and use them for versions of Schur’s Lemma corresponding to stronglyA couB pled re- ducibility, Theorems 5.1 and 5.2. Section 6 deals with a refinement of Schur’s Lemma for sets of normal matrices and corresponding versions for pairs , which are coupled normal, Theorem 6.2. The Appendix presents examplesA B to support some claims made in Section 3. 2 Reducibility and Schur’s Lemma We begin by reviewing the ordinary notion of reducibility for a set of linear transformations and Schur’s Lemma. Definition 2.1. A set, , of linear transformations, on an n-dimensional T vector space, , is reducible if there is a proper, non-zero subspace of such that T ( V) for all T . If is not reducible, we sayU itV is irreducible. U ⊆ U ∈ T T The subspace is an invariant subspace for each transformation T in . We say is fullyU reducible if it is possible to decompose as a direct sumT T V = ˆ, where and ˆ are both nonzero, proper invariant subspaces ofV .U ⊕ U U U TAlternatively, one can state this in matrix terms. The linear transforma- tions in may be represented as n n matrices, relative to a choice of basis for . LetT d be the dimension of ;× choose a basis for in which the first d basisV vectors are a basis for . SinceU is an invariantV subspace of each T U U in , the matrices representing relative to this basis are block upper tri- angularT with square diagonal blocksT of sizes d d and (n d) (n d).
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